1: Congruence

1.1: Experiment with transformations in the plane.

KY.HS.G.1: Know and apply precise definitions of the language of Geometry:

Circles
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Inscribed Angles
Parallel, Intersecting, and Skew Lines

KY.HS.G.1.a: Understand properties of line segments, angles and circle.

Chords and Arcs
Circles
Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Inscribed Angles
Investigating Angle Theorems
Parallel, Intersecting, and Skew Lines
Segment and Angle Bisectors

KY.HS.G.1.b: Understand properties of and differences between perpendicular and parallel lines.

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Parallel, Intersecting, and Skew Lines

KY.HS.G.2: Representing transformations in the plane.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

KY.HS.G.2.a: Describe transformations as functions that take points in the plane as inputs and give other points as outputs.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

KY.HS.G.2.b: Compare transformations that preserve distance and angle measures to those that do not.

Dilations
Reflections
Rotations, Reflections, and Translations
Translations

KY.HS.G.2.c: Given a rectangle, parallelogram, trapezoid, or regular polygon, formally describe the rotations and reflections that carry it onto itself, using properties of these figures.

Reflections
Rotations, Reflections, and Translations
Similar Figures

KY.HS.G.3: Develop formal definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines, parallel lines and line segments.

Circles
Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

KY.HS.G.4: Understand the effects of transformations of geometric figures.

KY.HS.G.4.a: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

KY.HS.G.4.b: Specify a sequence of transformations that will carry a given figure onto another.

Dilations
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

KY.HS.G.4.c: Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure. Given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Absolute Value with Linear Functions
Circles
Dilations
Holiday Snowflake Designer
Reflections
Rotations, Reflections, and Translations
Similar Figures
Translations

1.2: Understand congruence in terms of rigid motions.

KY.HS.G.5: Know and apply the concepts of triangle congruence:

KY.HS.G.5.a: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Proving Triangles Congruent

KY.HS.G.5.b: Explain how the criteria for triangle congruence (ASA, SAS and SSS) follow from the definition of congruence in terms of rigid motions.

Proving Triangles Congruent

1.3: Prove geometric theorems.

KY.HS.G.6: Apply theorems for lines, angles, triangles, parallelograms.

Constructing Parallel and Perpendicular Lines
Investigating Angle Theorems
Isosceles and Equilateral Triangles
Parallelogram Conditions
Polygon Angle Sum
Special Parallelograms
Triangle Angle Sum
Triangle Inequalities

KY.HS.G.7: Prove theorems about geometric figures.

KY.HS.G.7.a: Construct formal proofs to justify theorems for lines, angles and triangles.

Congruence in Right Triangles
Investigating Angle Theorems
Isosceles and Equilateral Triangles
Triangle Angle Sum
Triangle Inequalities

KY.HS.G.7.b: Construct formal proofs to justify theorems for parallelograms.

Parallelogram Conditions
Special Parallelograms

1.4: Make geometric constructions.

KY.HS.G.8: Create and apply geometric constructions.

KY.HS.G.8.a: Make formal geometric constructions with a variety of tools and methods.

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Segment and Angle Bisectors

KY.HS.G.8.b: Apply basic construction procedures to construct more complex figures.

Constructing Congruent Segments and Angles
Constructing Parallel and Perpendicular Lines
Segment and Angle Bisectors

2: Similarity, Right Triangles and Trigonometry

2.1: Understand similarity in terms of similarity transformations.

KY.HS.G.9: Understand properties of dilations.

KY.HS.G.9.a: Verify the properties that result from that dilations given by a center and a scale factor.

Dilations
Similar Figures

KY.HS.G.9.b: Verify that a dilation produces an image that is similar to the pre-image.

Circles
Dilations
Similar Figures

KY.HS.G.10: Apply the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Proving Triangles Congruent
Similar Figures

2.2: Prove theorems involving similarity.

KY.HS.G.11: Understand theorems about triangles.

KY.HS.G.11.a: Apply theorems about triangles.

Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures
Similarity in Right Triangles

KY.HS.G.11.b: Prove theorems about triangles.

Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Similar Figures
Similarity in Right Triangles

KY.HS.G.11.c: Use similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Chords and Arcs
Congruence in Right Triangles
Constructing Congruent Segments and Angles
Dilations
Perimeters and Areas of Similar Figures
Proving Triangles Congruent
Similar Figures
Similarity in Right Triangles

2.3: Define trigonometric ratios and solve problems involving right triangles.

KY.HS.G.12: Understand properties of right triangles.

KY.HS.G.12.a: Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles (sine, cosine and tangent).

Sine, Cosine, and Tangent Ratios

KY.HS.G.12.b: Explain and use the relationship between the sine and cosine of complementary angles.

Cosine Function
Sine Function

KY.HS.G.12.c: Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Cosine Function
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard
Sine Function
Sine, Cosine, and Tangent Ratios
Tangent Function

3: Circles

3.1: Understand and apply theorems about circles.

KY.HS.G.15: Verify using dilations that all circles are similar.

Circles

KY.HS.G.16: Identify and describe relationships among angles and segments within the context of circles involving:

KY.HS.G.16.a: Recognize differences between and properties of inscribed, central and circumscribed angles.

Chords and Arcs
Inscribed Angles

KY.HS.G.16.b: Understand relationships between inscribed angles and the diameter of a circle.

Circumference and Area of Circles
Inscribed Angles

KY.HS.G.17: Apply basic construction procedures within the context of a circle.

KY.HS.G.17.a: Construct the inscribed and circumscribed circles of a triangle.

Concurrent Lines, Medians, and Altitudes

3.2: Find arc lengths and areas of sectors of circles.

KY.HS.G.18: Understand the relationship between an intercepted arc length within a circle and the radius of the circle.

KY.HS.G.18.a: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius. Derive the formula for the area of a sector.

Chords and Arcs

KY.HS.G.18.b: Define the radian measure of the angle as the measure of a central angle that intercepts an arc equal in length to the radius of the circle.

Cosine Function
Sine Function
Tangent Function

4: Expressing Geometric Properties with Equations

4.1: Translate between the geometric description and the equation for a conic section.

KY.HS.G.19: Understand the relationship between the algebraic form and the geometric representation of a circle.

KY.HS.G.19.a: Write the equation of a circle of given center and radius using the Pythagorean Theorem.

Circles

KY.HS.G.19.b: Derive and write the equation of a circle of given center and radius using the Pythagorean Theorem.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

KY.HS.G.19.c: Complete the square to find the center and radius of a circle given by an equation.

Circles
Distance Formula
Pythagorean Theorem
Pythagorean Theorem with a Geoboard

KY.HS.G.20: Derive the equations of conic sections.

KY.HS.G.20.a: Derive the equation of a parabola given a focus and directrix.

Parabolas

KY.HS.G.20.b: Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Ellipses
Hyperbolas

4.2: Use coordinates to prove simple geometric theorems algebraically.

KY.HS.G.22: Justify and apply the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)

KY.HS.G.23: Find measurements among points within the coordinate plane.

KY.HS.G.23.a: Use points from the coordinate plane to find the coordinates of a midpoint of a line segment and the distance between the endpoints of a line segment.

Distance Formula

KY.HS.G.24: Use coordinates within the coordinate plane to calculate measurements of two dimensional figures.

Distance Formula

KY.HS.G.24.a: Compute the perimeters of various polygons.

Distance Formula

KY.HS.G.24.b: Compute the areas of triangles, rectangles and other quadrilaterals.

Distance Formula

5: Geometric Measurement and Dimensions

5.1: Explain volume formulas and use them to solve problems.

KY.HS.G.25: Analyze and determine the validity of arguments for the formulas for the various figures and shapes.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

KY.HS.G.25.a: Finding the circumference and area of a circle.

Circumference and Area of Circles
Prisms and Cylinders
Pyramids and Cones

KY.HS.G.25.b: Finding the volume of a sphere, prism, cylinder, pyramid and cone.

Prisms and Cylinders
Pyramids and Cones

KY.HS.G.26: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

Prisms and Cylinders
Pyramids and Cones

KY.HS.G.27: Use volume formulas to solve problems for cylinders, pyramids, cones, spheres, prisms.

Prisms and Cylinders
Pyramids and Cones

5.2: Visualize relationships between two-dimensional and three-dimensional objects.

KY.HS.G.28: Identify the shapes of two-dimensional cross-sections of three-dimensional objects and identify three-dimensional objects generated by rotations of two-dimensional objects.

Pyramids and Cones

6: Modeling with Geometry

6.1: Apply geometric concepts in modeling situations.

KY.HS.G.29: Use geometric shapes, their measures and their properties to describe objects in real world settings.

Prisms and Cylinders
Pyramids and Cones

KY.HS.G.30: Apply concepts of density based on area and volume in modeling situations, using appropriate units of measurement.

Prisms and Cylinders